This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# Extension

 Extensionis an option for various polynomial and algebraic functions that specifies generators for the algebraic number field to be used.
• For polynomial functions, Extension determines the algebraic number field in which the coefficients are assumed to lie.
• The setting Extension->a specifies the field consisting of the rationals extended by the algebraic number a.
• The must be exact numbers, and can involve radicals as well as Root and AlgebraicNumber objects.
• Extension specifies that any algebraic numbers that appear in the input should be included in the extension field.
• For polynomial functions, the default setting Extension->None specifies that all coefficients are required to be rational. Any algebraic numbers appearing in input are treated like independent variables.
• Extension includes both the and any algebraic numbers in the input.
Factor a polynomial over :
PolynomialGCD over the field generated by the algebraic numbers present in the coefficients:
Factor a polynomial over :
 Out[1]=

PolynomialGCD over the field generated by the algebraic numbers present in the coefficients:
 Out[1]=
 Scope   (8)
By default, factorization is performed over the rationals:
This specifies the factorization should be done over the rationals extended by :
Here the factorization is done over the rationals extended by and I:
By default, PolynomialGCD treats algebraic numbers as independent variables:
This computes the GCD over the algebraic number field generated by the coefficients:
By default, Together treats algebraic numbers as independent variables:
With Extension, Together recognizes algebraically dependent coefficients:
By default, the norm is computed in the field generated by the AlgebraicNumber object:
This computes the norm in the field in which the AlgebraicNumber object is represented:
This computes the norm in the field generated by :
For Factor, Extension->I is equivalent to GaussianIntegers->True: